More importantly, the general right to property is not the same thing at all as the specific, concrete right to use and/or dispose of a particular item or resource. In other words, the right to property does not give any particular person title to any specific property--it just means that an individual has the right to acquire title to property (provided the previous owner is willing to transfer title,) and that having title to property gives the owner the right to decide how, when, where and by whome said property will be used.
Similary, the right to keep and bear arms does not mean that the government (or anyone else) has to provide you with at least one gun and one bullet, free of charge. Nor does it mean that you have the right to get or stay hired by an employer, regardless of your politics, religion, sexual orientation or gun ownership status.
You have the right to own a gun. An employer has the right to deny you employment for any reason whatsoever--or for no reason at all. A right confers freedom of action within some defined scope or domain, but it stops where that domain ends (as defined by title or its equivalent,) or where the specific rights of others (as defined by title or its equivalent) would be violated. The key concept is title, which is necessary to define which particular items or resources are actually owned by which parties.
As a practical matter, this is simply untrue. Even more so once employment has been awarded.
If employers property rights confer to them the authority to dictate the contents of an employees means of private conveyance, they similarly face forfeiture if an employees means of private conveyance contains illegal drugs. Employers face no such risk of forfeiture, therefore the contents of an employees means of private conveyance are beyond the employers reach.